Finite Group Databasebeta

← #79156

Subgroups — #79156

Order 384.

356 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 125 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5511C212yesmaximal
2C2×C3×#107C211yesmaximal, characteristic, complemented
3#5116C211yesmaximal, characteristic, complemented
4C8×#67C211yesmaximal, characteristic
5#4970C212yesmaximal, complemented
6#4887C211yesmaximal, characteristic, complemented
7#4884C212yesmaximal, complemented
8D3×#107C211yesmaximal, characteristic, complemented
9#4875C212yesmaximal, complemented
10#4874C211yesmaximal, characteristic, complemented
11#4873C211yesmaximal, characteristic, complemented
12#3338—31nomaximal, Sylow
13C22×C3×C8C2211yescharacteristic
14C2×C3×C4⋊C4C2212yes—
15C4×#67—22no—
16C4×#67C2211yescharacteristic
17#924—22no—
18#924C2212yes—
19C2×Dic12C2211yescharacteristic
20C2×D3×C8C2211yescharacteristic
21#897C2212yes—
22#896—22no—
23D3×C4⋊C4C2212yes—
24#889—22no—
25#888C2212yes—
26#887C2212yes—
27C3×#107C2211yescharacteristic
28C3×#107C2211yescharacteristic
29C3×#107C2212yes—
30#830C2211yescharacteristic
31(C2×D3)⋊C8C2211yescharacteristic
32#818C2211yescharacteristic
33#816C2212yes—
34Dic3⋊C8C2211yescharacteristic
35C8×Dic3C2211yescharacteristic
36#810C2212yes—
37#810C2212yes—
38#807C2212yes—
39#807C2212yes—
40#554—32no—
41#500—31no—
42#484—32no—
43#484—32no—
44#477—32no—
45#467—31no—
46#467—31no—
47C4×Q16—31no—
48C8×D4—31no—
49C2×#107—31no—
50C2×#107D311yescharacteristic, complemented
51C22×C3×C4—22no—
52C22×C3×C4C2311yescharacteristic
53C2×#67D411yescharacteristic
54C2×C4×D3—22no—
55C2×C4×D3C2311yescharacteristic
56C2×Dic6—22no—
57C2×Dic6C2312yes—
58C2×C3×C8—22no—
59C2×C3×C8C2311yescharacteristic
60C2×C3×C8C2311yescharacteristic
61C3×C4⋊C4—22no—
62C3×C4⋊C4C2312yes—
63C3×C4⋊C4C2312yes—
64#217—22no—
65#217D411yescharacteristic
66(C2×D3)⋊C4—22no—
67(C2×D3)⋊C4D411yescharacteristic
68C4⋊Dic3—22no—
69C4⋊Dic3C2312yes—
70Dic3⋊C4—22no—
71Dic3⋊C4—22no—
72Dic3⋊C4—22no—
73Dic3⋊C4—22no—
74Dic3⋊C4D411yescharacteristic
75C4×Dic3—22no—
76C4×Dic3C2311yescharacteristic
77C2×C3⋊C8C2311yescharacteristic
78C2×C3⋊C8C2311yescharacteristic
79Dic12—22no—
80D3×C8—22no—
81C2×Q16—31no—
82C22×C8—31no—
83C22×C8C2×D311yescharacteristic
84C4⋊Q8—32no—
85#125—62no—
86C22⋊Q8—32no—
87C22⋊Q8—32no—
88C22⋊Q8—62no—
89C22⋊Q8—62no—
90C4×Q8—32no—
91C4×D4—31no—
92C4×D4—62no—
93C2×C4⋊C4—32no—
94C2×C4⋊C4C2×D312yes—
95#107—31no—
96#107—32no—
97#107—32no—
98#107C2×D311yescharacteristic
99#107C2×D311yescharacteristic
100#107C2×D312yes—
101C4⋊C8—31no—
102#103—32no—
103#103—32no—
104#103—32no—
105C22⋊C8—31no—
106C22⋊C8—31no—
107C4×C8—31no—
108C23×C3C2×D411yescharacteristic
109C22×D3C2×D411yescharacteristic
110C2×C3×C4—21no—
111C2×C3×C4—21no—
112C2×C3×C4—22no—
113C2×C3×C4—22no—
114C2×C3×C4—82no—
115C2×C3×C4C2×D411yescharacteristic
116C2×C3×C4C2411yescharacteristic
117#67Q1614yescomplemented
118C2×Dic3—22no—
119C2×Dic3—22no—
120C2×Dic3C2×D411yescharacteristic
121C2×Dic3C2×D411yescharacteristic
122C2×Dic3C2×D411yescharacteristic
123C4×D3—21no—
124C4×D3—21no—
125C4×D3—82no—
126Dic6—22no—
127Dic6—22no—
128Dic6—82no—
129C3×C8—21no—
130C3×C8#4112yes—
131C3⋊C8—21no—
132C3⋊C8—21no—
133C2×Q8—32no—
134C2×Q8—62no—
135C2×D4—31no—
136C22×C4—22no—
137C22×C4—31no—
138C22×C4—62no—
139C22×C4C22×D311yescharacteristic
140Q16—62no—
141C2×C8—22no—
142C2×C8—31no—
143C2×C8—31no—
144C2×C8—62no—
145C2×C8C22×D311yescharacteristic
146C2×C8C22×D311yescharacteristic
147C4⋊C4—22no—
148C4⋊C4—31no—
149C4⋊C4—32no—
150C4⋊C4—62no—
151C4⋊C4—62no—
152C4⋊C4—62no—
153C4⋊C4—62no—
154C4⋊C4—62no—
155C4⋊C4C22×D312yes—
156C4⋊C4C22×D312yes—
157C22⋊C4—31no—
158C22⋊C4—31no—
159C22⋊C4—62no—
160C22⋊C4—62no—
161C42—31no—
162C42—62no—
163C22×C3—21no—
164C22×C3—21no—
165C22×C3C2×Q1612yes—
166C22×C3C22×D411yescharacteristic
167C2×D3—21no—
168C2×D3—21no—
169C2×D3C2×Q1612yes—
170C3×C4—21no—
171C3×C4—42no—
172C3×C4—42no—
173C3×C4C2×#4111yescharacteristic
174C3×C42−1+411yescharacteristic
175Dic3—21no—
176Dic3—21no—
177Dic3—42no—
178Dic3—42no—
179Dic3C2×Q1612yes—
180C23—31no—
181C23D3×D411yescharacteristic
182Q8—62no—
183Q8—62no—
184Q8—242no—
185D4—34no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—31no—
191C2×C4—31no—
192C2×C4—31no—
193C2×C4—61no—
194C2×C4—61no—
195C2×C4—62no—
196C2×C4—62no—
197C2×C4—82no—
198C2×C4—242no—
199C2×C4D3×D411yescharacteristic
200C2×C4C23×D311yescharacteristic
201C8—21no—
202C8—61no—
203C8—61no—
204C8#23512yes—
205C2×C3—22no—
206C2×C3#54711yescharacteristic
207C2×C3C22×Q1611yescharacteristic
208C2×C3#57611yescharacteristic
209D3—22no—
210C22—21no—
211C22—21no—
212C22—32no—
213C22—61no—
214C22—61no—
215C22D3×Q1612yes—
216C22C2×D3×D411yescharacteristic
217C4—21no—
218C4—32no—
219C4—42no—
220C4—42no—
221C4—61no—
222C4—61no—
223C4—122no—
224C4—122no—
225C4C2×#23511yescharacteristic
226C4#100711yescharacteristic
227C3#333811yescharacteristic, complemented, Sylow
228C2—22no—
229C2—62no—
230C2#550711yescharacteristic
231C2#575711yescharacteristic
232C2C2×D3×Q1611yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC2×C3×#107C2
socleC22×C3C22×D4
prime radicalC2×#67D4
hypercenterC2×#107D3
hypocenterC3#3338
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#107

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×C3×#107C2semidirect
3. #5116C2semidirect
5. #4970C2semidirect
6. #4887C2semidirect
7. #4884C2semidirect
8. D3×#107C2semidirect
9. #4875C2semidirect
10. #4874C2semidirect
11. #4873C2semidirect
50. C2×#107D3semidirect
117. #67Q16semidirect
227. C3#3338semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#55112
C2×C3×#1072
#51162
C8×#672
#49702
#48872
#48842
D3×#1072
#48752
#48742
#48732
#33383